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doi: 10.3934/naco.2021018
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Analysis of Rayleigh Taylor instability in nanofluids with rotation

1. 

Research Scholar, Deparment of Mathematics, NIILM University, Kaithal, Haryana, India

2. 

Department of Mathematics, Post Graduate Government College, Sector-11, Chandigarh, India

3. 

Department of Mathematics, NIILM University, Kaithal, Haryana, India

* Corresponding author: jyotiahuja1985@gmail.com

Received  December 2020 Revised  April 2021 Early access May 2021

This article focuses on the hidden insights about the Rayleigh-Taylor instability of two superimposed horizontal layers of nanofluids having different densities in the presence of rotation factor. Conservation equations are subjected to linear perturbations and further analyzed by using the Normal Mode technique. A dispersion relation incorporating the effects of surface tension, Atwood number, rotation factor and volume fraction of nanoparticles is obtained. Using Routh-Hurtwitz criterion the stable and unstable modes of Rayleigh-Taylor instability are discussed in the presence/absence of nanoparticles and presented through graphs. It is observed that in the absence/presence of nanoparticles, surface tension helps to stabilize the system and Atwood number has a destabilizing impact without the consideration of rotation factor. But if rotation parameter is considered (in the absence/presence of nanoparticles) then surface tension destabilizes the system while Atwood number has a stabilization effect (for a particular range of wave number). The volume fraction of nanoparticles destabilizes the system in the absence of rotation but in the presence of rotation the stability of the system is significantly stimulated by the nanoparticles.

Citation: Pooja Girotra, Jyoti Ahuja, Dinesh Verma. Analysis of Rayleigh Taylor instability in nanofluids with rotation. Numerical Algebra, Control and Optimization, doi: 10.3934/naco.2021018
References:
[1]

J. Ahuja and U. Gupta, Magneto convection in rotating nanofluid layer: Local thermal non-equilibrium model, American Scientific Publisher, 8 (2019), 1-9. 

[2]

J. Ahuja and P. Girotra, Analytical and numerical investigation of Rayleigh-Taylor instability in nanofluids, Pramana - J. Phys., 95 (2021), 25.

[3]

K. A. Baldwin, M. M. Scase and R. J. A. Hill, The inhibition of Rayleigh Taylor instability by rotation, Sci. Rep., 5 (2015), 11706.

[4]

B. S. Bhaduria, A. Kumar, J. Kumar, N. C. Sacheti and P. Chandran, Natural convection in a rotating anisotropic porous layer with internal heat generation, Transport in Porous Media, 90 (2011), 687–705. doi: 10.1007/s11242-011-9811-0.

[5]

P. K. Bhatia, Rayleigh-Taylor instability of a viscous compressible plasma of variable density, Astrophysics and Space Science, 26 (1974), 319-325. 

[6]

J. Buongiorno, Convective transport in nanofluids, ASME J. Heat Transf., 128 (2006), 240-250. 

[7]

B. B. Chakraborthy, A note on Rayleigh Taylor insability in presence of rotation, Z. Angew. Math. Mech., 59 (1959), 651-652. 

[8]

B. B. Chakraborthy, Hydromagnetic Rayligh Taylor instabilityof rotating stratified fluid, Phys. Fluids, 25 (1982), 743-747.  doi: 10.1063/1.863828.

[9]

S. Chandrasekhar, The character of the equilibrium of an incompressible heavy viscous fluid of variable density, Proc. Cambridge Philos. Soc., 51 (1955), 162–178. doi: 10.1017/s0305004100030048.

[10]

S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, 2$^{nd}$ edition, Dover Publication, New York, 1981.

[11]

S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, In Development and Applications of Non-Newtonian flows, FED- 231/MD(eds. D. A. Siginer and H. P. Wang), ASME, 66 (1955), 99-105.

[12]

S. K. DasN. PutraP. Thiesen and W. Roetzel, Temperature dependence of thermal conductivity enhancement for nanofluids, ASME J. Heat Transfer, 25 (2003), 567-574. 

[13]

L. A. Dávalos-Orzoco, Rayleigh-Taylor instability of a continuously stratified fluid under a general rotation field, Phys. Fluids A, 1 (1989), 1192-1199. 

[14]

N. F. El-AnsaryG. A. HoshoudyA. S. Abd-Elrady and A. H. A. Ayyad, Effects of surface tension and rotation on Rayleigh Taylor instability, Phys. Chem. Chem. Phys., 4 (2002), 1464-1470. 

[15]

R. Hide, The character of the equilibrium of a heavy, viscous, incompressible, rotating fluid of variable density ii. two special cases, Q. J. Mech. Appl. Math., 9 (1956), 35-50.  doi: 10.1093/qjmam/9.1.35.

[16]

P. Kumar, Rayleigh Taylor instability of rotating Oloroydian viscoelastic fluids in porous medium in presence of a variable magnetic field, Jnanabha, 24 (1977), 127-134. 

[17]

H. MasudaA. EbataK. Teramae and N. Hishinuma, Alteration of thermal conductivity and viscosity of liquid by dispersing ultra fine particles, Netsu Bussei, 7 (1993), 227-233. 

[18]

D. A. Nield and A. V. Kuznetsov, Thermal instability in a porous medium layer saturated by nanofluid, Int. J. Heat Mass Transfer, 52 (2008), 5796-5801.  doi: 10.1007/s11242-009-9413-2.

[19]

L. Rayleigh, Investigation of the character of equilibrium of an incompressible heavy fluid of variable density, Proc. Roy. Math. Soc., 14 (1883), 170-177.  doi: 10.1112/plms/s1-14.1.170.

[20]

P. K. Sahrma, A. Tiwari and S. Argal, Effect of magnetic field on the Rayleigh Taylor instability of rotating and stratified plasma, IOP Conf. Series: Journal of Physics: Conf. Series, 836 (2017), 012009.

[21]

M. M. Scase, K. A. Baldwin and R. J. A. Hill, The rotating Rayleigh Taylor instability, Physical Review Fluids, 2 (2017), 024801 (1-21).

[22]

R. C. Sharma, MHD instability of rotating superposed fluids through porous medium, Acta Physica Academiae Scientiarium Hungaricace, 42 (1977), 21-28.  doi: 10.1007/BF03157196.

[23]

R. C. SharmaP. Kumar and S. Sharma, Rayleigh Taylor instability of Rivlin-Ericksen elastico-viscous fluid through porous medium, Indian Journal of Physics, 4 (2001), 337-340. 

[24]

R. C. Sharma and V. K. Bhardwaj, Rayleigh Taylor instability of Newtonian and Oldroydian viscoelastic fluids in porous medium, Z. Naturforsch, 49a (1994), 927-930. 

[25]

J. J. Tao, X. T. He, W. H. Ye and F. H. Busse, Nonlinear Rayleigh Taylor instability of inviscid fluids, Physical Review E, 87 (2013), 013001 (1-4).

[26]

G. I. Taylor, The instability of liquid surfaces when accelerated in a direction perpendicular to their planes, Proc. R. Soc. Ser. A., 201 (1950), 192-196.  doi: 10.1098/rspa.1950.0052.

[27]

D. Y. Tzou, Thermal instability of nanofluids in natural convection, Int. J. of Heat Mass Transf., 51 (2008), 2967-2979. 

[28]

A. Volk and C. J. Khaler, Density model for aqueous glycerol solutions, Experiments in Fluids, 59 (2018), 75 (1-4).

show all references

References:
[1]

J. Ahuja and U. Gupta, Magneto convection in rotating nanofluid layer: Local thermal non-equilibrium model, American Scientific Publisher, 8 (2019), 1-9. 

[2]

J. Ahuja and P. Girotra, Analytical and numerical investigation of Rayleigh-Taylor instability in nanofluids, Pramana - J. Phys., 95 (2021), 25.

[3]

K. A. Baldwin, M. M. Scase and R. J. A. Hill, The inhibition of Rayleigh Taylor instability by rotation, Sci. Rep., 5 (2015), 11706.

[4]

B. S. Bhaduria, A. Kumar, J. Kumar, N. C. Sacheti and P. Chandran, Natural convection in a rotating anisotropic porous layer with internal heat generation, Transport in Porous Media, 90 (2011), 687–705. doi: 10.1007/s11242-011-9811-0.

[5]

P. K. Bhatia, Rayleigh-Taylor instability of a viscous compressible plasma of variable density, Astrophysics and Space Science, 26 (1974), 319-325. 

[6]

J. Buongiorno, Convective transport in nanofluids, ASME J. Heat Transf., 128 (2006), 240-250. 

[7]

B. B. Chakraborthy, A note on Rayleigh Taylor insability in presence of rotation, Z. Angew. Math. Mech., 59 (1959), 651-652. 

[8]

B. B. Chakraborthy, Hydromagnetic Rayligh Taylor instabilityof rotating stratified fluid, Phys. Fluids, 25 (1982), 743-747.  doi: 10.1063/1.863828.

[9]

S. Chandrasekhar, The character of the equilibrium of an incompressible heavy viscous fluid of variable density, Proc. Cambridge Philos. Soc., 51 (1955), 162–178. doi: 10.1017/s0305004100030048.

[10]

S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, 2$^{nd}$ edition, Dover Publication, New York, 1981.

[11]

S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, In Development and Applications of Non-Newtonian flows, FED- 231/MD(eds. D. A. Siginer and H. P. Wang), ASME, 66 (1955), 99-105.

[12]

S. K. DasN. PutraP. Thiesen and W. Roetzel, Temperature dependence of thermal conductivity enhancement for nanofluids, ASME J. Heat Transfer, 25 (2003), 567-574. 

[13]

L. A. Dávalos-Orzoco, Rayleigh-Taylor instability of a continuously stratified fluid under a general rotation field, Phys. Fluids A, 1 (1989), 1192-1199. 

[14]

N. F. El-AnsaryG. A. HoshoudyA. S. Abd-Elrady and A. H. A. Ayyad, Effects of surface tension and rotation on Rayleigh Taylor instability, Phys. Chem. Chem. Phys., 4 (2002), 1464-1470. 

[15]

R. Hide, The character of the equilibrium of a heavy, viscous, incompressible, rotating fluid of variable density ii. two special cases, Q. J. Mech. Appl. Math., 9 (1956), 35-50.  doi: 10.1093/qjmam/9.1.35.

[16]

P. Kumar, Rayleigh Taylor instability of rotating Oloroydian viscoelastic fluids in porous medium in presence of a variable magnetic field, Jnanabha, 24 (1977), 127-134. 

[17]

H. MasudaA. EbataK. Teramae and N. Hishinuma, Alteration of thermal conductivity and viscosity of liquid by dispersing ultra fine particles, Netsu Bussei, 7 (1993), 227-233. 

[18]

D. A. Nield and A. V. Kuznetsov, Thermal instability in a porous medium layer saturated by nanofluid, Int. J. Heat Mass Transfer, 52 (2008), 5796-5801.  doi: 10.1007/s11242-009-9413-2.

[19]

L. Rayleigh, Investigation of the character of equilibrium of an incompressible heavy fluid of variable density, Proc. Roy. Math. Soc., 14 (1883), 170-177.  doi: 10.1112/plms/s1-14.1.170.

[20]

P. K. Sahrma, A. Tiwari and S. Argal, Effect of magnetic field on the Rayleigh Taylor instability of rotating and stratified plasma, IOP Conf. Series: Journal of Physics: Conf. Series, 836 (2017), 012009.

[21]

M. M. Scase, K. A. Baldwin and R. J. A. Hill, The rotating Rayleigh Taylor instability, Physical Review Fluids, 2 (2017), 024801 (1-21).

[22]

R. C. Sharma, MHD instability of rotating superposed fluids through porous medium, Acta Physica Academiae Scientiarium Hungaricace, 42 (1977), 21-28.  doi: 10.1007/BF03157196.

[23]

R. C. SharmaP. Kumar and S. Sharma, Rayleigh Taylor instability of Rivlin-Ericksen elastico-viscous fluid through porous medium, Indian Journal of Physics, 4 (2001), 337-340. 

[24]

R. C. Sharma and V. K. Bhardwaj, Rayleigh Taylor instability of Newtonian and Oldroydian viscoelastic fluids in porous medium, Z. Naturforsch, 49a (1994), 927-930. 

[25]

J. J. Tao, X. T. He, W. H. Ye and F. H. Busse, Nonlinear Rayleigh Taylor instability of inviscid fluids, Physical Review E, 87 (2013), 013001 (1-4).

[26]

G. I. Taylor, The instability of liquid surfaces when accelerated in a direction perpendicular to their planes, Proc. R. Soc. Ser. A., 201 (1950), 192-196.  doi: 10.1098/rspa.1950.0052.

[27]

D. Y. Tzou, Thermal instability of nanofluids in natural convection, Int. J. of Heat Mass Transf., 51 (2008), 2967-2979. 

[28]

A. Volk and C. J. Khaler, Density model for aqueous glycerol solutions, Experiments in Fluids, 59 (2018), 75 (1-4).

Figure 1.  Effect of surface tension on growth rate parameter of Rayleigh Taylor instability in the absence of nanoparticles with rotation and without rotation for varying T = 0.1, 0.2
Figure 2.  Effect of Atwood number on growth rate parameter of Rayleigh Taylor instability in the absence of nanoparticles with rotation and without rotation for varying A = 0.11, 0.119
Figure 3.  Effect of rotation parameter on growth rate parameter of Rayleigh Taylor instability in the absence of nanoparticles with rotation and without rotation for varying $ \Omega $ = 0.1, 0.2, 0.3
Figure 4.  Effect of surface tension on growth rate parameter of Rayleigh Taylor instability in the presence of same kind of nanoparticles with rotation and without rotation for varying T = 0.0001, 0.0002
Figure 5.  Effect of Atwood number on growth rate parameter of Rayleigh Taylor instability in the presence of same kind of nanoparticles with rotation and without rotation for varying A = 0.11, 0.119
Figure 6.  Effect of volume fraction of nanoparicles on growth rate parameter of Rayleigh Taylor instability in the presence of same kind of nanoparticles with rotation and without rotation for varying $ \phi $ = 0.001, 0.005
Figure 7.  Effect of rotation number on growth rate parameter of Rayleigh Taylor instability in the presence of same kind of nanoparticles $ \Omega $ = 0.1, 0.2, 0.3
Figure 8.  Effect of surface tension on growth rate parameter of Rayleigh Taylor instability in the presence of different kind of nanoparticles with rotation and without rotation for varying T = 0.0001, 0.0002
Figure 9.  Effect of Atwood number on growth rate parameter of Rayleigh Taylor instability in the presence of same kind of nanoparticles with rotation and without rotation for varying A = 0.11, 0.119
Figure 10.  Effect of volume fraction of nanoparicles on growth rate parameter of Rayleigh Taylor instability in the presence of different kind of nanoparticles with rotation and without rotation for varying $ \phi $ = 0.001, 0.005
Figure 11.  Effect of rotation number on growth rate parameter of Rayleigh Taylor instability in the presence of diffrent kind of nanoparticles $ \Omega $ = 0.1, 0.2, 0.3
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